have a gravity profile given by,
\(g=-{g}_{e}e^{-\cfrac{x}{{r}_{e}}}+{g}_{m}e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}\)
We have also seen that integrating this expression with respect to x give us Gravitational Potential Energy, GPE,
\(GPE=-\int{g}dx=\int{+{g}_{e}e^{-\cfrac{x}{{r}_{e}}}-{g}_{m}e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}}dx\)
\(GPE=C-{g}_{e}{r}_{e}e^{-\cfrac{x}{{r}_{e}}}-{g}_{m}{r}_{m}e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}\)
If GPE=0 when x=0,
\( C={ g }_{ e }{ r }_{ e }+{ g }_{ m }{ r }_{ m }e^{ -\cfrac { Orbs }{ { r }_{ m } } }\)
\(\cfrac{d GPE(Orbs)}{d Orbs}=-{ g }_{ m }e^{ -\cfrac { Orbs }{ { r }_{ m } } }+{g}_{m}e^{(-\cfrac{Orbs}{{r}_{m}}+\cfrac{x}{{r}_{m}})}\)
Which is a positive term for all values of Orbs. That means it will take energy to increase the value of Orbs; that energy input is required to pull the bodies further apart.
And from the graph of gravity along the line joining the two centers of the bodies, we see that gravity is pointing away, to greater distance apart, at the end points. In the case of the Moon and Earth, gravity at the Moon side is pointing away from Earth. The Moon has a tendency to fall in the direction of gravity which is away from Earth. It will not fall to Earth, simply because gravity is outwards on the Moon side. And it will not drift way unless energy is supplied to the system.
And that's why the Moon can only fall in love.
